Conjectures on representations involving primes

TL;DR

The paper introduces 100 new conjectures related to prime representations, aiming to inspire number theorists and stimulate further research into prime-related number decompositions and properties.

Contribution

It presents a large set of novel conjectures involving primes and related functions, expanding the scope of prime representation problems.

Findings

Proposes 100 new conjectures on prime representations.

Provides specific examples and formulations involving primes and rational numbers.

Aims to motivate future research in prime number theory.

Abstract

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists $k\in\{0,\ldots,n\}$ such that $n+k$ and $n+k^2$ are both prime. (ii) Each integer $n>1$ can be written as $x+y$ with $x,y\in\{1,2,3,\ldots\}$ such that $x+ny$ and $x^2+ny^2$ are both prime. (iii) For any rational number $r>0$, there are distinct primes $q_1,\ldots,q_k$ with $r=\sum_{j=1}^k1/(q_j-1)$. (iv) Every $n=4,5,\ldots$ can be written as $p+q$, where $p$ is a prime with $p-1$ and $p+1$ both practical, and $q$ is either prime or practical. (v) Any positive rational number can be written as $m/n$, where $m$ and $n$ are positive integers with $p_m+p_n$ a square (or $\pi(m)\pi(n)$ a positive square), $p_k$ is the $k$-th prime and $\pi(x)$ is the prime-counting function.